I am tempted to believe that "closing the loop" is not a desirable course of action - the Care Loop forces Care to go through the NLM state, implied blatantly to be one of abject misery, as part of it- rather than forcing her to continue through this Loop, we need to bust out of it.

I think this is a very good point. Makes me realize that the way Care's description is worded is quite insidious when you read it carefully. It's easy to think it means "help Care NLM recover so she becomes Care A, and stop her from becoming NLM again", but 'closing the loop' seems to imply trapping her in the A/B/NLM cycle as you say. In various media, when someone is stuck in a loop, breaking out of it is the ideal outcome.

It leads me to wonder, is Care not stuck in a loop yet? Does she exist in her own timeline independently from the other events in the game?